Well seeing as I did literally the exact same thing as you, I'll give you all I know before either of us play, I've looked into each game slightly by the way :
-Hitman Codename 47, Not sure myself yet, but apparently contracts consists of remakes of this game, however some missions have not been remade so it's likely you should start here still.
-Hitman 2 : Silent Assassin, apparently for hardcore stealth fans but entirely worth your time, this game begins the actual development of Agent 47 and opens up the world more with things like his handler Diana.
-Hitman 3 : Contracts, Refer to Hitman 1, though as I understand it this game features an Agent 47 in the future reflecting on his past, for reasons I won't reveal.
-Hitman 4 : Bloodmoney, This game is rumoured to be as of now the best in series, offering something for Hardcore and casual stealth fans, there is apparently next to no story in this game until towards the later missions, and is mainly sold on it's fantastic re-playability and sandbox style maps offering multiple approaches.
-Hitman : Sniper Challenge, This is actually a prelude to Absolution and is essentially what it's described as, you cannot move, you are set up with a sniper and have to assassinate your targets from a number of ways, just imagine this as a cut level from absolution, as all's it was was a pre-order bonus prior to Absolution.
-Hitman 5 : Absolution, This being the only one I have completed on the hardest difficulty mind you and with no kills barring my targets !!! it offers various environments, a diffcult or easy experience with various ways to play such as stealth or guns blazing and a more emotional look into what is the normally cold Agent 47, now after seeing why people were against this in it's initial release, I cannot deny what people say; This game looks like it offers multiple paths in theory, but it's actually a rather linear game; Without spoiling anything please note that whilst there are many ways of playing this game, there is only one "perfect" way to play each mission.